© Denzil Green
There are two different versions of quesadillas; both involve melted cheese inside a tortilla.
In Tex-Mex cooking (American-style Mexican), a quesadilla is a tortilla on the bottom, cheese in the middle and a tortilla on top, making a rounded cheese “sandwich” which is then cooked and served on a plate. Both tortillas are flour tortillas.
In Mexico style, a single corn tortilla is used, folded over to make a half moon (though in Northern Mexico, closer to the American border, corn tortillas are replaced by flour tortillas, which has almost certainly influenced the American-style use of flour tortillas, too.) Mexican style is meant to be held in the hand and eaten.
Good cheeses to use for either version include Monterey Jack, goat cheese, Oaxaca Cheese, Queso Fresco, Queso Blanco, Manchego, Chihuahua (sic), Caciotta, or Cotija. A mild cheddar cheese makes a good substitute. You can mix cheeses; a good melting cheese like Monterey Jack (or mild cheddar), with a flavourful cheese such as Cotija (or Pecorino Romano.)
In addition to the cheese, other fillings can go inside. They can be anything — meat, vegetable, even fruit. There shouldn’t be too much, though, so that the Quesadilla becomes bulky and awkward. The fillings inside really won’t have any time to cook, just heat, so bear that in mind: meat and fish fillings should be precooked, as well as veg such as potatoes, etc.
To make, put a bit of oil in a skillet or griddle (expect corn tortillas to need a bit more oil than flour ones.) Cook on one side until golden brown, then flip and cook the other side until golden brown and the cheese has melted.